arXiv:hep-ph/0501289v2 30 May 2007
PITHA 05/01
hep-ph/0501289
31 January 2005
Third-order Coulomb corrections
to the S-wave Green function, energy levels and
wave functions at the origin
M. Beneke, Y. Kiyo, K. Schuller
Institut für Theoretische Physik E, RWTH Aachen,
D – 52056 Aachen, Germany
Abstract
We obtain analytic expressions for the third-order corrections due to the strong
interaction Coulomb potential to the S-wave Green function, energy levels and
wave functions at the origin for arbitrary principal quantum number n. Together
with the known non-Coulomb correction this results in the complete spectrum
of S-states up to order α5s . The numerical impact of these corrections on the
Upsilon spectrum and the top quark pair production cross section near threshold
is estimated.
1
Introduction
Several years ago advances in non-relativistic effective theory made it possible to compute
quarkonium properties at the next-to-next-to-leading order (NNLO). Since the perturbative approach assumes that the non-relativistic energy scale E ∼ mαs2 is larger than
the strong interaction scale ΛQCD , these computations apply to the lowest bottomonium
states and heavy quark current spectral functions near threshold such as those relevant
to top quark pair production. NNLO results have been obtained for matching coefficients [1, 2, 3], energy levels [4, 5, 6, 7], wave functions at the origin [5, 6, 7] and spectral
functions [6, 7, 8, 9, 10], principally for the S-wave states, which are the most important
ones for applications [11].
It was observed that the perturbative corrections are almost always very large. Although the origin of these large corrections can sometimes be understood as being due
to mass renormalization [12] or large logarithms, it is currently believed that a complete third (next-to-next-to-next-to-leading/NNNLO) order calculation is necessary to
describe accurately even the case of top quark pair production near threshold. That this
is a difficult undertaking is reflected by the fact that partial results at NNNLO exist for
various quantities [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], but only the S-wave ground
state energy is currently fully known at third order [24], save for an unknown constant
a3 in the Coulomb potential. In this paper, we take one further step and compute the
NNNLO corrections from the QCD Coulomb potential to the S-wave energy levels and
wave function at the origin for arbitrary principal quantum number n, and to the S-wave
Green function (spectral function) relevant to top quark pair production. Together with
the known non-Coulomb energy level corrections [18, 25] this determines the S-wave
energy levels completely at third order for any n. The correction to the Green function
and wave functions at the origin forms part of the complete third-order top quark pair
production cross section. Another motivation for first concentrating on the Coulomb
corrections is that the Schrödinger equation with the Coulomb potential can be solved
numerically, and the result can be compared to the perturbative computation. This is
no longer possible once the singular non-Coulomb potentials are included, in which case
the perturbative approach is the only option. Comparing the perturbative approximation to the numerical solution allows us to estimate the convergence of the successive
approximations.
2
Outline of the calculation
The computation of Coulomb corrections can be phrased in the language of elementary
quantum mechanics. We consider the Hamiltonian
H =−
∇2
+ V (r),
m
1
(1)
where m denotes the heavy quark pole mass, and V the Coulomb potential of the strong
force. With
Z
d3 q −iqx
e
Ṽ (q),
(2)
V (r) =
(2π)3
the potential reads, up to the fourth order in the expansion in the strong coupling αs ,
Ṽ (q) = −
4πCF αs
+ δ Ṽ (q),
q2
(3)
4πCF αs αs
µ2
αs
δ Ṽ (q) = −
a
+
β
ln
+
1
0
2
2
q
4π
q
4π
(
αs
+
4π
3 a3 + 8π 2 CA3 ln
+
3a1 β02
2 a2 + 2a1 β0 + β1
µ2
µ2
ln 2 + β02 ln2 2
q
q
µ2
ν2 +
3a
β
+
2a
β
+
β
ln 2
2
0
1
1
2
q2
q
µ2
µ2
5
+ β0 β1 ln2 2 + β03 ln3 2
2
q
q
)
.
(4)
Here βi are the coefficients of the QCD β-function in the MS-scheme, defined with the
P
convention ∂as /∂ ln µ2 = − βn an+2
, as ≡ αs /(4π) such that β0 = 11CA /3 − 4TF nf /3
s
[26]. The constants a1 , a2 are given in [3], the αs4 CA3 ln ν 2 /q 2 term is from [13], while
the three-loop constant term a3 , currently unknown, is estimated to be 6240 (nf = 4)
and 3840 (nf = 5) [27]. The group theory factors for SU(Nc ) gauge theory (Nc = 3 in
QCD) are CF = (Nc2 − 1)/(2Nc ), CA = Nc , TF = 1/2, and nf denotes the number of
quarks whose masses are smaller than mαs and neglected (nf = 4 for bottom systems,
nf = 5 for top). αs refers to the strong coupling in the MS scheme at the renormalization
scale µ, while ν is a scale necessary to define the separation of potential and ultrasoft
effects. The dependence on this factorization scale cancels in physical quantities when
the ultrasoft correction is included.
When δ Ṽ (q) = 0 the spectrum of the Hamiltonian reproduces the well-known Bohr
energy levels. This will be our zeroth order approximation. We consider quarkonium
systems in which αs (mαs ) is small, and compute the spectrum in a perturbation expansion in αs . Hence, the coefficient of αsn+1 in δ Ṽ (q) is considered a perturbation of the
nth order, and the third-order result we are aiming at requires up to three insertions of
the αs2 potential, but only one insertion of the order αs4 potential. We shall focus on the
S-wave Green function
G(E) ≡ h0|Ĝ(E)|0i = h0|
1
|0i,
H − E − iǫ
(5)
where |0i denotes a position eigenstate with eigenvalue r = 0, and compute the matrix
element of the right-hand side of (omitting the argument E of the Green function)
Ĝ = Ĝ0 − Ĝ0 δV1 Ĝ0 − Ĝ0 δV2 Ĝ0 + Ĝ0 δV1 Ĝ0 δV1 Ĝ0
− Ĝ0 δV3 Ĝ0 + 2Ĝ0 δV1 Ĝ0 δV2 Ĝ0 − Ĝ0 δV1 Ĝ0 δV1 Ĝ0 δV1 Ĝ0 .
2
(6)
G0 (E) denotes the zeroth-order Green function, and δVn the nth order perturbation
potential. The Green function G(E) has single poles at the exact S-wave energy levels
E = En , such that
|ψn (0)|2
E→E
.
(7)
G(E) = n
En − E − iǫ
Inserting
En(0)
En =
2
αs
αs
1 + e1 +
4π
4π
|ψn(0) (0)|2
|ψn (0)| =
2
αs
e2 +
4π
αs
αs
f1 +
1+
4π
4π
2
3
e3 + . . . ,
αs
f2 +
4π
!
3
(8)
!
f3 + . . .
(9)
into this equation, we can determine e1,2,3 and f1,2,3 by comparing the expansion of (7)
in αs with (6) near E = En . We recall that En(0) = −m(αs CF )2 /(4n2 ), and |ψn(0) (0)|2 =
(mαs CF )3 /(8πn3 ). The details of the calculation are too lengthy to be reproduced here.
However, we sketch the computation of the threefold insertion h0|Ĝ0δV1 Ĝ0 δV1 Ĝ0 δV1 Ĝ0 |0i
of δV1 in the Appendix.
There is an equivalent quantum-field-theoretical description of the calculation, which
is the appropriate one in the context of systematic higher-order computations of quarkonium properties [11]. After all the quantum-mechanical description breaks down beyond some accuracy, since (a) the system is sensitive to the short-distance quantum
fluctuations, and (b) the restriction to the quark-antiquark sector of the Fock space is
inadequate. Systematic calculations therefore start from QCD and obtain effective Lagrangians by systematically integrating out short-distance fluctuations (∆x ∼ 1/m) as
well as all massless degrees of freedom down to the ultrasoft energy scale E ∼ mαs2 ,
which characterizes the size of energy fluctuations in a quarkonium bound state. The
result is an effective Lagrangian, in which the potentials appear as matching coefficients
[28, 29]. In particular the Coulomb potential is regarded as the matching coefficient of
a relevant operator, and differs from the Wilson loop definition beginning at order αs4 .
Consequently, the αs4 ln αs term which appears in the static potential [30] is absent, and
replaced by a logarithm of the factorization scale ν [13] as shown in (4). The computation
of Coulomb corrections is based on the Lagrangian
Leff
∂2
∂2
= ψ (x) i∂ +
ψ(x) + χ† (x) i∂ 0 −
χ(x)
2m
2m
+
!
0
†
Z
h
!
i
h
i
d3 r ψ † ψ (x + r ) V (r) χ† χ (x),
(10)
where V (r) is the (colour-singlet) Coulomb potential, and where the non-relativistic twocomponent field ψ(x) (χ(x)) destroys (creates) a heavy quark (anti-quark). There are
other terms in the effective Lagrangian that have to be included for a complete NNNLO
calculation. The point we wish to make here is that these are well-understood, and hence
there is a well-defined procedure to separate the complete calculation into several simpler
3
parts, of which the calculation of Coulomb corrections is one. An important feature of
(10) is that the leading-order Coulomb potential cannot be treated as a perturbation,
and must be included in the zeroth order approximation together with the bilinear terms.
Ĝ0 is essentially the propagator of this theory. We can also obtain G(E) in (5) directly
by computing the two point function
Nc G(E) = i
Z
d4 x eiqx hΩ|T ([ψ † χ](x)[χ† ψ](0)|Ωi,
(11)
with |Ωi the Fock space vacuum state. This makes it clear that the Green function
is closely related to inclusive heavy quark-anti-quark cross sections in e− e+ collisions,
where the case of the top quark is particularly interesting.
3
Third-order corrections to bound state parameters
In this section we present our result for the third-order correction from the Coulomb
potential to the S-wave energy levels and wave functions at the origin (squared) for
arbitrary principal quantum number n. The corrections (notation as in (8,9)) are parameterized as
nC
ei = eC
i + ei ,
(12)
fi = fiC + finC ,
(13)
where C stands for the correction, when only the Coulomb potential is included in the
effective Lagrangian. By definition ‘nC’ denotes all the remaining corrections, which
originate from additional potentials and an ultrasoft non-potential interaction. Together
with the non-Coulomb third-order correction enC
to energy level [18, 25], we obtain
3
a complete result for the bound state energies to order mαs5 for any n. For the wave
function, however, f3nC is not yet known. We should emphasize that we define |ψn (0)|2 by
the residue of the correlation function (11) of non-relativistic currents. The non-Coulomb
corrections arise from potentials which cause short-distance singularities, such that finC
are factorization scheme-dependent quantities. This scheme-dependence is canceled by
short-distance coefficients, which we do not discuss here, but which are known to the
same order as finC is known (i ≤ 2).
3.1
Energy levels
The energy corrections from the Coulomb potential are organized as follows,
eC
1 = 4β0 L + cE,1 ,
2 2
eC
2 = 12β0 L + L
(14)
− 8β02 + 4β1 + 6β0 cE,1 + cE,2,
4
(15)
eC
3
=
32β03
+L
3
2
L +L
16β03
−
56β03
+ 28β0 β1 +
− 16β0 β1 + 4β2 −
+cE,3 + 32π
2
CA3
24β02 cE,1
12β02cE,1
+ 6β1 cE,1 + 8β0 cE,2
nν
+ S1 (n) ,
ln
mCF αs
(16)
where L = ln (nµ/(mCF αs )) and Sa (n) = nk=1 1/k a is the harmonic sum. For later
convenience we introduce the nested harmonic sums
P
n
X
1
Sa,b (n) ≡
S (k),
a b
k=1 k
Sa,b,c (n) ≡
n
X
1
S (k).
a b,c
k=1 k
(17)
In the following we omit the argument of harmonic sums which is always understood to
be the principal quantum number n. The coefficients of the logarithmic terms are fixed
by the renormalization group in terms of the β-function and the cE,i from lower orders.
The “non-trivial” information is encoded in the constants cE, i . The first and second
order corrections, cE, 1 , cE, 2 are known [4, 5, 6, 7]
cE, 1 = 2a1 + 4S1 β0 ,
(18)
cE, 2 = a21 + 2a2 + 4S1 β1 + 4a1 β0 3S1 − 1
+β02
"
S1
2π 2
8
+ 16S2 − 8nS3 +
12S1 − 8 −
+ 8nξ(3) ,
n
3
#
(19)
−s
where ξ(s) is the zeta-function, ξ(s) = ∞
k=1 k . Our new result is the third-order
correction to En for arbitrary n, which reads
P
h
i
h
i
h
cE, 3 = 2a1 a2 + 2a3 + 2a21 β0 4S1 − 5 + 4a2 β0 4S1 − 1 + 4a1 β1 3S1 − 1
"
+4S1 β2 + β0 β1 S1
+a1 β02
+β03
+S2
"
"
S1
i
7π 2
24
+ 36S2 − 16nS3 +
+ 16nξ(3)
28S1 − 16 −
n
3
32
8π 2
48S1 − 56 −
+ 64S2 − 32nS3 + 8 +
+ 32nξ(3)
n
3
S1 S1
#
#
16 32π 2
32
+ 96S2 − 64nS3 + 16 +
+
+ 64nξ(3)
32S1 − 56 −
n
n
3
!
16 40nπ 2
−
− 16n2 ξ(3) + S3 96 + 16n + 8n2 π 2
8nS2 + 16n S3 − 32 −
n
3
!
2
!
4π 2 2nπ 4
−104nS4 + 48n S5 − 144S2,1 + 224nS3,1 − 32n S3,2 − 96n S4,1 −
+
3
45
2
2
2 2
+ξ(3) 32 − 16n − 8n π
#
2
+ 96n ξ(5) .
5
2
(20)
For completeness we also give the non-Coulomb correction. The first-order correction
is only from the Coulomb potential, thus enC
= 0. The second-order term was first
1
obtained in [4], the third-order term for arbitrary n in [25]. The expressions are


~2
CF 2 
11
2S
CA CF
2
,
+
2
−
−
enC
/(16π
)
=
2
n
n
16n
3
(21)
49nf TF CA CF
4CA 2 CF 97
2
enC
/(64π
)
=
−
+
− ln 2 + ln n + ln(CF αs ) − S1
3
36n
3n
48
"
5
CA 3
− − ln 2 − 2 ln n + 4 ln (CF αs ) − 3 ln(ν/m) − 2S1
+
6
6
"
+CA CF 2
"
#
#
~2
107 7 ln n 7S1 7 ln(CF αs )
1 139 7 ln n
7
S
−
+
+
−
−
+
+
n
108
6
6
6
12n
n 36
6
!
47
2
41 ln(CF αs ) 7S1
−
− 4 ln 2 + 2 − + ln 2 − ln n − ln(CF αs ) − S1
+
6
6
3n
16
CF 3
~ 2 − 79 + 7 − 8 ln 2 + 7 ln n − 7S1 + 9 ln (CF αs )
− 2nLE (n) + S
+
3n
6
2n
"
#
#
~2
CF 2 nf TF
5
TF CF 2 32
10 S
~ 2 ln 2 − 1
+
−8+
+
+
− 2 ln 2 + S
9n
2n
3
n
15
"
"
+β0 CA CF
#
"
π2
1
2
L−
+
+ S2 + C F 2
n
6
2n
2
~ 2 π + 1 − 1 − 2S2
+S
9
2n 6n2
3
!
#
~ 2
11
4 4S
− 2+ −
L
8n
n
3n
11S1
1
3
π2
−
+
2S
+
+
−
2
8n2
n 8 n2
3
!#
a1 CA CF
9
C 2 a1
7 ~2
+
−
,
+ F
+ −S
2n
2n
8n 2
"
#
(22)
~ 2 is the eigenvalue of the spin operator. For the spin-triplet (singlet) state
where S
~ 2 = 2 (S
~ 2 = 0). (The Coulomb potential is spin-independent, hence the eC do not
S
i
~ 2 .) Furthermore LE (n) denotes the “Bethe logarithm”, which must be
depend on S
evaluated numerically. For n = 1, 2, . . . we find
LE (n) = (−81.5379, −37.671, −22.4818, −14.5326, −9.52642, −6.0222, . . .) .
(23)
The first three numbers have been obtained in [14]. We have performed an independent
calculation of the ultrasoft correction.
6
3.2
Wave functions at the origin
The wave function corrections fi from the Coulomb potential are given by
f1C = 6β0 L + cψ,1 ,
f2C = 24β02 L2 + L
f3C
=
80β03
3
(24)
2
L +L
− 12β02 + 6β1 + 8β0 cψ,1 + cψ,2 ,
−
108β03
+ 54β0 β1 +
40β02 cψ,1
(25)
+L 24β03 − 24β0 β1 + 6β2 − 16β02 cψ,1 + 8β1 cψ,1 + 10β0 cψ,2 + cψ,3
2
+48π CA
3
1
nπ 2
nν
+
S1 + 2nS2 − 1 −
ln
mCF αs
3
3
!
.
(26)
The first and second order corrections are known [5, 7]
cψ, 1 = 3a1 + 2β0
3a21
cψ, 2 =
"
nπ 2
S1 + 2nS2 − 1 −
,
3
#
+ 3a2 + 2a1 β0
"
4nπ 2
4S1 + 8nS2 − 7 −
3
12 8nπ 2
−
8S1 + 16nS2 − 20 −
n
3
"
+β02 S1
(27)
!
+ S2
#
+ 2β1
"
nπ 2
S1 + 2nS2 − 1 −
3
4n2 π 2
4n2 S2 + 8 − 8n −
3
#
!
(3 + 4n)π 2 n2 π 4
+28nS3 − 20n S4 − 24nS2,1 + 16n S3,1 + 4 +
+
+ 20nξ(3) . (28)
3
9
2
#
2
Our new result for the third-order correction to |ψn (0)|2 for arbitrary n is
cψ, 3 = a31 + 6a1 a2 + 3a3 + 10a21 β0
8 nπ 2
− −
5
3
"
+β0 β1 S1
#
+ 8a1 β1
"
"
31 nπ 2
−
S1 + 2nS2 −
10
3
7 nπ 2
S1 + 2nS2 − −
4
3
#
36 20nπ 2
−
22S1 + 40nS2 − 44 −
n
3
+ 2β2
!
"
#
"
+ 10a2 β0 S1 + 2nS2
nπ 2
S1 + 2nS2 − 1 −
3
+ S2
#
8n2 π 2
8n S2 + 14 − 16n −
3
2
(21 + 16n)π 2 2n2 π 4
+
+ 48nξ(3)
+64nS3 − 40n2 S4 − 56nS2,1 + 32n2 S3,1 + 8 +
6
9
+a1 β02
"
S1
60 40nπ 2
−
40S1 + 80nS2 − 116 −
n
3
!
+ S2
!
#
20n2 π 2
20n S2 + 40 − 72n −
3
2
5n2 π 4
+140nS3 − 100n S4 − 120nS2,1 + 80n S3,1 + 48 + (5 + 12n)π +
+ 100nξ(3)
9
2
2
2
7
!
#
+β03
"
S1 4S1 4S1 + 16nS2 − 19 −
6 8nπ 2 + 8S2 3n2 S2 + 2 − 14n − n2 π 2
−
n
3
+104nS3 − 120n2 S4 − 112nS2,1 + 96n2 S3,1 + 80 +
64 (58 + 56n)π 2 2n2 π 4
+
+
n
3
3
!
+120nξ(3) + S2 − 4n(17 + 2n)S2 + 72n2 S3 − 96n2 S2,1 + 64n3 S3,1 − 96 + 16n
24 8(5 − n)nπ 2
− −
− 8n2 ξ(3) + S3 − 16n3 S3 + 64 − 16n − 20n2 π 2 + 32n3 ξ(3)
n
3
!
+S4
64n3 π 2
68n + 40n +
3
−32S3,1
2
n3 π 2
15n
+ n2 +
2
3
!
!
− 312n S5 + 144n S6 + S2,1 48n − 120 + 16n π
!
+ 384n2 S3,2 + 576n2 S4,1 − 224n3 S4,2 − 256n3 S5,1
2
3
2 2
!
8(2 + n)π 2
+256nS2,1,1 + 64n S2,2,1 − 64n S2,3,1 − 448n S3,1,1 + 192n S4,1,1 − 8 −
3
2
3
2
3
(83 + 10n)nπ 4 4n3 π 6
+
+ ξ(3) 48 − 80n − 12n2 π 2 − 16n3 ξ(3) − 40n2 ξ(5) . (29)
−
45
105
!
#
The first-order non-Coulomb correction vanishes, f1nC = 0, as for the energy-levels.
Starting from second order |ψn (0)|2 depends on the factorization scheme that separates
the hard (relativistic) corrections from the low-energy corrections reproduced by the nonrelativistic effective theory. The second-order term was first obtained in [5] for the spin~ 2 = 2), but the result given there does not refer to the conventional MS
triplet states (S
subtraction scheme. The result of [5] including the hard correction has been reproduced
in [7], where the MS scheme was used for the individual contributions. Using these
results (not printed in [7]) we find for the S-wave spin-triplet non-Coulomb correction
to the wave function at the origin squared in the MS subtraction scheme
f2nC (µh )|MS /(16π 2)
=
CF2
"
15
4
22 2
2
L(µh ) − 2 +
+
− S1
3
8n
3n
9
3
+ CF CA
"
2 5
L(µh ) + + − S1
n 4
#
#
~ 2 = 2),
(S
(30)
where L(µf ) = ln (nµf /(mCF αs )) and µf refers to the factorization scale. The µf
dependence cancels in the product C 2 |ψn (0)|2 , where C denotes the hard matching
coefficient of the operator ψ † σ i χ [1, 2]. (σ i are the Pauli matrices.) The third-order
coefficient f3nC is unknown.
8
4
Quarkonium masses
In this section we compare the calculated energy levels with the bottomonium Υ(nS)
masses. We also discuss the masses of the would-be toponium states, which are relevant
to the top quark pair production cross section in high-energy e− e+ collisions.
First we give a numerical version of the general result for the S-state energy levels
~ 2 = 2), and nf = 4 (bottomonium) and nf = 5 (toponium).
for the spin-triplet case (S
For the first three states n = 1, 2, 3 we obtain, for nf = 4,
MΥ(1S)
"
4
= 2mb − mb αs2 1 + 3.590 + 2.653 L αs + 15.56 + 3.963nC + 12.07 L
9
+5.277 L2 αs2 + 76.35 + 6.289 â3 + [28.47 + 15.30 ln αs + 21.02 L]nC
2
3
+72.65 L + 27.59 L + 9.332 L
MΥ(2S)
"
αs3
,
(31)
1
= 2mb − mb αs2 1 + 4.916 + 2.653 L αs + 25.38 + 2.287nC + 17.34 L
9
2
+5.277 L
2
αs + 140.7 + 6.289â3 + [11.25 + 8.647 ln αs + 12.13 L]nC
+120.3 L + 41.59 L2 + 9.332 L3 αs3
MΥ(3S)
#
"
#
,
(32)
4
= 2mb − mb αs2 1 + 5.800 + 2.653 L αs + 32.90 + 1.593nC + 20.86 L
81
+5.277 L2 αs2 + 196.0 + 6.289 â3 + [4.559 + 6.305 ln αs + 8.449 L]nC
2
3
+157.3 L + 50.92 L + 9.332 L
αs3
#
.
(33)
Here mb denotes the bottom quark pole mass, L = ln(nµ/(mb CF αs (µ))), and we normalize the contribution from the unknown third-order constant in the Coulomb potential,
a3 , to the Padé estimate by defining â3 = a3 /a3, P ade . We have given the Coulomb and
non-Coulomb corrections separately to emphasize the numerical dominance of the former
(in the pole scheme). The quarkonium masses are of course independent of the ultrasoft
factorization scale ν, but the separation into a Coulomb and non-Coulomb correction is
not. The representations of the series above is given for ν = mb CF αs (µ)/n. We note
that the Coulomb correction increases with n, while the non-Coulomb corrections become smaller. The reason for this is that the characteristic distance scale (the “Bohr
radius”) increases hrn i ∼ n, hence the relative effect of the short-range non-Coulomb interactions decreases for the excited states. The third-order result for n = 1 has already
been obtained in [24], the other results are new.
9
For the spin-triplet toponium, nf = 5, the series read
Mtt̄(1S)
"
4
= 2mt − mt αs2 1 + 3.201 + 2.440 L αs + 12.47 + 3.963nC + 9.718 L
9
h
i
+4.467 L2 αs2 + 56.54 + 3.870 â3 + 26.85 + 15.30 ln αs + 19.34 L
#
nC
+52.88 L + 20.06 L2 + 7.267 L3 αs3 ,
Mtt̄(2S)
"
(34)
1
= 2mt − mt αs2 1 + 4.421 + 2.440 L αs + 20.54 + 2.287nC + 14.18 L
9
2
+4.467 L
αs2
h
i
+ 104.2 + 3.870 â3 + 10.35 + 8.647 ln αs + 11.16 L
#
nC
+88.59 L + 30.96 L2 + 7.267 L3 αs3 ,
Mtt̄(3S)
"
(35)
4
= 2mt − mt αs2 1 + 5.234 + 2.440 L αs + 26.74 + 1.593nC + 17.16 L
81
2
+4.467 L
αs2
h
i
+ 145.4 + 3.870 â3 + 3.934 + 6.305 ln αs + 7.773 L
2
3
+116.4 L + 38.23 L + 7.267 L
#
αs3 ,
nC
(36)
where mt is top quark pole mass.
The series coefficients are large and the series do not converge for bottomonium,
presumably because the pole mass introduces a strong infrared renormalon divergence
[31, 32], which is not present in the physical observable “quarkonium mass” itself [12, 33].
It is therefore an advantage to use a better mass renormalization convention such as the
potential-subtracted (PS) mass [12]. The relation to the pole mass appropriate to thirdorder calculations is given by
1
m = mPS (µf ) −
2
Z
q≤µf
d3 q
Ṽ (q)|ν=µf
(2π)3
"
µf CF αs
αs αs
= mPS (µf ) +
1+
2β0 l1 + a1 +
π
4π
4π
αs
+
4π
3 2 4β02 l2
+ 2 2a1 β0 + β1 l1 + a2
5
8β03l3 + 4 3a1 β02 + β0 β1 l2 + 2 3a2 β0 + 2a1 β1 + β2 l1
2
+ a3 + 16π
2
CA3
#
,
(37)
where l1 = ln(µ/µf ) + 1, l2 = ln2 (µ/µf ) + 2 ln(µ/µf ) + 2, l3 = ln3 (µ/µf ) + 3 ln2 (µ/µf ) +
10
6 ln(µ/µf ) + 6. Note that the PS mass is defined with the choice ν = µf in the Coulomb
potential Ṽ (q). The scale µf should be chosen of order mαs in order not to violate the
power counting of the non-relativistic expansion, so the relation (37) is accurate to order
mαs5 just as the third-order bound state masses. Our standard choice is µf = 2 GeV for
bottom and µf = 20 GeV for top.
4.1
Bottomonium
Before extracting quark masses and predicting bottomonium masses it is instructive
to display the convergence of the expansions at the “natural” renormalization scale µ,
where L = ln(nµ/(Mb CF αs (µ))) = 0. (Here Mb refers to the bottom quark mass in the
(nf =4)
chosen scheme.) For ΛQCD
= 290.4 MeV, and with 4-loop running of αs the “natural”
scale is realized at µ = (2.02, 1.30, 1.03) GeV (for n = 1, 2, 3) with mb = 5 GeV, and
µ = (1.91, 1.23, 0.98) GeV with mb,PS (2 GeV) = 4.6 GeV. For the numerical analysis we
adopt a3 = a3,P ade = 6270 (â3 = 1). Eqs. (31-33) show that the precise value of a3 is not
important as long as it is not very different from the Padé estimate.
Using the relation (37) between the pole and PS mass we re-express MΥ(nS) in terms
of mb,PS taking into account that µf /mb counts as one power of αs . With mb = 5 GeV,
mb,PS ≡ mb,PS (2 GeV) = 4.6 GeV we obtain
MΥ(1S) = 2mb +
(0)
E1
1 + 1.09NLO + 1.42 + 0.36nC
(0)
= 2mb,PS + E1,PS 1 + 0.19NLO + 0.07 − 0.23nC
(0)
MΥ(2S) = 2mb + E2
= 2mb,PS +
(0)
E2,PS
MΥ(3S) = 2mb +
= 2mb,PS +
1 + 0.26NLO + 0.26 − 0.05nC
1 + 0.25NLO + 0.41 − 0.03nC
with
N3 LO
+ 20.10 − 0.03nC
+ 0.69 + 0.00nC
, (38)
N3 LO
N3 LO
N3 LO
+ 0.37 − 0.03nC
+ 8.64 + 0.18nC
N2 LO
N2 LO
+ 2.29 + 0.28nC
N2 LO
+ 0.09 − 0.19nC
N2 LO
1 + 2.69NLO + 7.06 + 0.34nC
N2 LO
N2 LO
1 + 1.91NLO + 3.84 + 0.35nC
(0)
E3
(0)
E3,PS
, (39)
N3 LO
N3 LO
, (40)
(αs CF )2 mb,PS 2µf CF αs
+
.
(41)
4n2
π
This clearly shows that the series expansions are useless in the pole scheme, but the
successive terms are (slowly) decreasing in the PS scheme for n = 1. For n > 1 we still
observe that the transition to the PS scheme eliminates the huge correction from the
Coulomb potential present in the pole scheme, yet the series coefficients are no longer
converging. This is perhaps not surprising, because the scales are near or below 1 GeV
for n > 1, and a perturbative treatment is simply no longer justified.
(0)
En,PS = −
11
m PS(2GeV) from M(1 ) = 9:460[GeV℄
b;
4.7
S
NNLO
4.6
4.5
PSfrag replaements
LO
NNNLO
NLO
4.4
1
1.5
2
2.5
3
3.5
4
[GeV℄
Figure 1: The bottom PS mass, mb,PS (2 GeV), extracted from the experimental value
MΥ(1S) = 9.460 GeV as a function of renormalization scale µ at LO (long dashes, black),
NLO (long-short dashes, red), NNLO (short dashes, green) and NNNLO (solid, blue).
We can therefore use the experimental Υ(1S) mass MΥ(1S) = 9.460 GeV to extract
the bottom PS mass at NNNLO as was done in [34] at NNLO. In Figure 1 we show
the extracted PS mass as a function of the renormalization scale µ at LO (long dashes,
black), NLO (long-short dashes, red), NNLO (short dashes, green) and NNNLO (solid,
blue). Varying µ from 1.25 GeV to 4 GeV (as done in [34]) the NNNLO correction is
never larger than about 30 MeV and the error from the scale dependence is of similar size.
We therefore assign a ±30 MeV error to mb,PS from the truncation of the perturbative
expansion. The uncertainty in αs (MZ ) = 0.118 ± 0.003 results in a ±10 MeV error on
mb,PS . The largest uncertainty in the determination of the bottom quark mass from the
Υ(1S) mass is then from non-perturbative effects. The perturbative approximation is
justified when the ultrasoft scale mb (CF αs )2 ≫ ΛQCD , in which case the leading nonperturbative contributions is expressed in terms of the gluon condensate [35, 36]
np
δMΥ(1S)
=
624πmb hαs GGi
.
425 (mb CF αs )4
(42)
The numerical estimate is strongly dependent on the choice of scale in αs in the denominator. Referring to [34] for a more detailed discussion of the non-perturbative correction,
we obtain
mb,PS (2 GeV) = (4.57 ± 0.03pert. ± 0.01αs ± 0.07non−pert. ) GeV
(43)
as the final result of this analysis. Because of the small third-order correction the bottom
quark mass remains practically unchanged compared to the second-order analysis of
[34], and so does the MS mass determined from (43). Further improvement of the mass
12
PSfrag replaements
M(2S )
11
10.5
M(3S )
11
NNNLO
10.5
NNNLO
Exp
Exp
10
10
9.5
9
9.5
LO
1
1.5
2
2.5
[GeV℄
PSfrag replaements
3
3.5
9
4
LO
1
1.5
2
2.5
[GeV℄
3
3.5
4
Figure 2: Predicted masses of the Υ(2S) and Υ(3S) as a function of the renormalization
scale µ. The lines refer to LO (long dashes, black), NLO (long-short dashes, red),
NNLO (short dashes, green) and NNNLO (solid, blue). The widths of the bands for the
experimental mass values are exaggerated.
determination by this method requires a quantitative understanding of non-perturbative
effects.
Having determined mb,PS , we are in the position to predict the masses of the excited Slevel states at the third order. (An analysis of the complete spectrum at second order was
performed in [37].) We focus on the spin-triplet states Υ(2S) and Υ(3S). The successive
approximations up to the third order are shown in Figure 2 for mb,PS = 4.57 GeV as
a function of the renormalization scale. For µ between 2 GeV and 4 GeV it appears
that the large third-order correction is welcome to bring the theoretical result closer to
the observed masses. However, at the natural scales 1.23 GeV (n = 2) and 0.98 GeV
(n = 3) the NNLO result agrees well with data and the NNNLO correction renders
the prediction too large. As is apparent from the figure, the conclusion is that the
perturbative computation of bottomonium masses is applicable only to the ground state,
n = 1, while the excited states, involving larger distances, appear to be in the nonperturbative regime. It can be seen from (39,40) that the NNNLO term for n > 1 is
dominated by the Coulomb correction.
4.2
Toponium
We briefly discuss the situation for the toponium masses. Toponium bound states do
not exist in nature due to large decay width Γt ∼ 1.5 GeV of the top quark [38], however
the remnant of the 1S toponium state should be visible as an enhancement in the cross
section e− e+ → tt̄X near threshold. The convergence of the series expansion for the
toponium 1S mass is therefore a good measure for the accuracy to which the top quark
mass might be determined from this cross section [10].
The method suggested in [7, 29] relies on determining mt,PS ≡ mt,PS (20 GeV) from the
cross section measurement and obtaining the top quark MS mass from mt,PS , since both
13
relations are expected to be expressible in terms of well-behaved perturbative expansions.
(nf =5)
Adopting mt,PS = 175 GeV, nf = 5, ΛQCD
= 208 MeV and the natural scale µ =
32.6 GeV, where L = 0, we obtain
Mtt̄(1S) = (350 + 0.85 + 0.05 − 0.13 + 0.01) GeV = 350.78 GeV.
(44)
The sum of the series varies only by about 60 MeV when the scale is taken between 15
and 60 GeV, although the convergence is no longer satisfactory at the lower scale. The
small uncertainty implies that mt,PS can be determined with little theoretical error from
the cross section. The largest uncertainty in the determination of the top quark MS
mass then results from the unknown four-loop coefficient in the relation between the MS
mass and the pole mass, which is needed to convert mt,PS to the MS mass, as already
observed in [7]. This uncertainty is estimated to be around 100 MeV (see Table 2 of
[10]). Our conclusions are in good agreement with [39], where one of us investigated
the direct determination of the top quark MS mass from Mtt̄(1S) also using the NNNLO
result for the 1S energy level. We should emphasize that none of these estimates take into
account electroweak corrections, which are non-negligible, and must be included in the
mass relations and cross section prediction before a comparison with the experimental
cross section can be attempted.
5
Third-order Coulomb wave functions at the origin
and Green function
In this section we turn to the discussion of the S-wave Coulomb Green function, and
to the wave function of the origin squared (residues of the Green function at the bound
state poles). Since the third-order correction is not completely known, we include in
this section only the Coulomb corrections as defined in Section 3, i.e. we also neglect the
(known) non-Coulomb correction at second order. The series expansions seem to be out
of control for the wave functions in the bottomonium system, hence we focus on the case
of the top quark and set nf = 5. We also set a3 = a3,P ade = 3840.
5.1
Wave function at origin squared
The numerical version of the general result for the S-state Coulomb wave function at
the origin squared reads, for n = 1, 2, 3,
(mt CF αs )3
1 + αs − 0.4333 + 3.661 L + αs2 5.832 − 5.112 L + 8.933 L2
8π
ν
(45)
+ 39.72 L − 22.91 L2 + 18.17 L3 ,
+ αs3 − 13.73 + 6.446 ln
mt CF αs
|ψ1 (0)|2C =
|ψ2 (0)|2C
(mt CF αs )3
=
1 + αs − 0.1769 + 3.661 L + αs2 10.19 − 3.861 L + 8.933 L2
64π
14
j
1.4
1
(0)j2=j
(0)
1
(0)j2B for tt(1S )
1.2
NNNLO
1
NNLO
NLO
0.8
LO
PSfrag replaements
0.6
20
40
60
[GeV℄
80
100
Figure 3: The Coulomb wave function at the origin squared for the ground state (n = 1)
(0)
normalized by |Ψ1 (0)|2 at µB = 32.6 GeV is shown as a function of the renormalization
scale µ. The input parameters are mt,PS (20 GeV) = 175 GeV, ν = mt,PS CF αs (µ). The
lines refer to LO (long dashes, black), NLO (long-short dashes, red), NNLO (short dashes,
green) and NNNLO (solid, blue).
+ αs3
2ν
+ 65.31 L − 19.09 L2 + 18.17 L3 ,
− 20.36 + 6.446 ln
mt CF αs
(46)
(mt CF αs )3
=
1 + αs 0.07953 + 3.661 L + αs2 13.27 − 2.609 L + 8.933 L2
216π
3ν
3
2
3
+ αs − 22.86 + 6.446 ln
(47)
+ 83.07 L − 15.27 L + 18.17 L .
mt CF αs
|ψ3 (0)|2C
We show the successive approximations for n = 1 in Figure 3. As before, we reexpressed the expansion in terms of the PS mass, treating µf ∼ mt αs . However, we note
that contrary to the energy levels, the introduction of the PS mass does not change qualitatively the behaviour of the expansion, neither would this be expected on theoretical
grounds. Our reference top quark mass is mt,PS (20 GeV) = 175 GeV, and the ultrasoft
factorization scale is chosen to be ν = mt,PS CF αs (µ) such that the corresponding logarithm in (45) vanishes. We also normalized the result to the LO wave function at the
scale µB ≡ mt,PS CF αs (µB ) = 32.6 GeV. It is clearly seen from the figure that the approximations converge, and that the inclusion of the new third-order correction stabilizes the
prediction further, provided µ is larger than about 25 GeV. We find a similar behaviour
for n = 2, 3, where, however, the enhancement of the wave function relative to leading
order is about 50% (n = 2) and 100% (n = 3) rather than roughly 5% as at n = 1.
It may be disconcerting that the perturbative expansion breaks down already at
scales as large as 20 GeV, where the strong coupling is still small. A more detailed
analysis shows that this early breakdown is caused primarily by the αs (αs β0 ln q 2 )n terms
15
in the Coulomb potential. We also note that at µ = µB , where L = 0, the series
expansions shown above are sign-alternating in contrast to the corresponding expressions
for the energy levels, which exhibit fixed-sign behaviour. It is a general fact that for
sign-alternating series of a certain regularity, the convergence of the expansion is much
improved by choosing a larger scale [40], since this renders the series coefficients and the
expansion parameter αs smaller. This explains the stability seen in the figure towards
scales larger than the natural scale µ = µB , and suggests that an error estimate for
|ψn (0)|2 from varying the scale between µB /2 and 2µB may be misleadingly large. We
shall see in the following subsection that this is indeed the case.
5.2
Green function
In addition to the energy levels and wave functions at the origin we have also computed
the full S-wave Green function up to the third order in the presence of the Coulomb
potential as described in Section 2. The result is expressed in terms of multiple sums
that can be evaluated numerically only.
The Coulomb Green function plays an important role in the calculation of inclusive
top quark pair production e− e+ → tt̄X near threshold [41], since the bulk of the cross
section is given by
R=
18πe2t
σttX
(1 + aZ ) Im G(E + iΓt )
=
σµ+ µ−
m2t
(48)
where et = 2/3 is the top quark electric charge, Γt the top quark width, and aZ accounts
for the vector coupling to the Z boson. The convergence of the perturbative approximation up to the second order has been the subject of many investigations several years
ago (see the review [10]).
We are now in the position to extend this investigation to the third order as far as the
corrections from the Coulomb potential are concerned. We computed the perturbative
expansion of the Green function given in (6). This strict expansion is never a good
approximation, because it contains terms of the form
"
αs En(0)
(0)
En − (E + iΓt )
#k
,
(49)
which originate from the expansion of (7) around En(0) rather than the true pole position.
These terms become numerically large near E ≈ En(0) , but they can be summed by adding
the exact pole structure and subtracting the expanded structure to the appropriate order,
see [7] for the corresponding expressions. In the following, “perturbative approximation”
means that this resummation is included. Alternatively, we computed the Green function
(5) numerically by solving the Schrödinger equation with the Coulomb potential (4) exactly, following the method described in [42]. We shall refer to this as the “exact result”.
The exact result contains an arbitrary number of insertions of the perturbation potentials
16
δV1 , δV2 , δV3 . The leading difference to the third-order perturbative approximation consists of fourth-order terms of the form Ĝ0 δV1 Ĝ0 δV1 Ĝ0 δV1 Ĝ0 δV1 Ĝ0 , Ĝ0 δV1 Ĝ0 δV1 Ĝ0 δV2 Ĝ0
etc. Comparing the two results we obtain an estimate of the importance of these multiple
insertions and of the convergence of the perturbative approximation.
(nf =5)
For the following numerical study we assume ΛQCD
= 0.208 GeV (αs (MZ ) = 0.118),
four-loop evolution of αs , Γt = 1.5GeV, and ν = 20 GeV. It is crucial for an accurate
prediction of the cross section not to use the top quark pole mass as an input parameter
[7, 10]. Our result is presented in terms of the top-quark potential-subtracted mass
mt,PS (20 GeV) = 175 GeV defined in (37). The conversion to the top quark MS mass
is given in [10]. We implement the PS scheme by working with an order-dependent
pole mass. That is, in the Nk LO perturbative approximation we compute the top quark
(k)
pole mass mt from mt,PS (20 GeV) = 175 GeV according to (37) including the terms
(k)
up to order µf αsk+1 , and use mt as an input to the Green function. For instance, for
(k)
renormalization scale µ = 30 GeV we obtain mt = (176.21, 176.56, 176.74, 176.87) GeV
for k = 0, 1, 2, 3. The energy argument of the Green function is then
√
(k)
E = s − 2mt
(50)
√
with s the center-of-mass energy of the e− e+ collision.
The upper plot in Figure 4 shows the convergence of the successive perturbative
approximations to the exact result for the Green function (cross section) at the renormalization scale µ = 30 GeV. The location of the “peak” position is indeed stable under
the inclusion of higher-order corrections as expected in the PS scheme. On the other
hand, the corrections to the magnitude of the cross section near the peak are significant,
decreasing from about 20% at NLO to about 7% at NNNLO. The corrections alternate
in sign as expected from the behaviour of the series expansion of |ψ1 (0)|2. An important
observation is that the third-order perturbative approximation coincides with the exact
result within 1%. Hence the higher-order insertions of the perturbation potentials are
negligible. The convergence of the approximations then suggests that the residual error
from yet higher perturbation Coulomb potentials δV4 etc. should be less than 5%.
In the lower plot of Figure 4 we display the renormalization scale dependence of the
third-order perturbative approximation (solid lines, µ = (60, 30, 15) GeV from top to
bottom). It is immediately apparent that the scale dependence is very small from 30
to 60 GeV, but the result for µ = 15 GeV is far away. We can trace this anomalous
behaviour directly to the breakdown of the perturbative expansion for |ψ1 (0)|2 at scales
below 25 GeV discussed in the previous subsection and displayed in Figure 3. The exact
result (dashed line) does not exhibit this behaviour for µ = 15 GeV, hence we conclude
that the multiple insertions of the perturbation potentials become large at small scales
and destroy the agreement of the perturbative and exact result. Indeed, we find that the
series of multiple insertions is very slowly converging at small scales. We therefore learn
the important lesson that the “correct” choice of scale in the perturbative approach is
µ > 25 GeV, while choosing smaller scales would lead to misleadingly large uncertainties.
The lower plot of Figure 4 indicates that the scale dependence is less than 5%, similar
17
1.2
1.0
R 0.8
LO
NLO
0.6
NNLO
NNNLO
NNNLO exact
349
350
351
√
s [GeV]
352
353
1.0
R
0.8
0.6
µ = 15 GeV (lower)
µ = 30 GeV (middle)
µ = 60 GeV (upper)
0.4
349
µ = 15 GeV exact
350
351
√
s [GeV]
352
353
Figure 4: Top quark pair production cross section (Coulomb corrections only) for mt,PS =
175 GeV, Γt = 1.5 GeV. Upper panel: successive approximations up to the third order
for µ = 30 GeV. Lower panel: Scale dependence of the third-order approximation. See
text for further explanation.
18
in size to the truncation error estimated above. This discussion does not include the
unknown third-order non-Coulomb corrections, but it lends support to the optimistic
interpretation that the magnitude of the tt̄ threshold cross section can eventually be
computed with an accuracy of a few percent.
6
Conclusion
We computed the third-order corrections from the strong interaction Coulomb potential
to the energy levels and the wave functions at the origin of the S-waves bound states, and
to the expectation value of the S-wave Green function operator in the |r = 0i state. We
view this as a first step towards a complete calculation of the third-order top quark pair
production cross section e− e+ → tt̄X near threshold. It also completes the expression
for the S-wave quarkonium masses with accuracy mαs5 for arbitrary principal quantum
number n, except for the unknown constant a3 in the Coulomb potential.
We updated the determination of the bottom quark mass from the mass of the Υ(1S)
and obtain
mb,PS (2 GeV) = (4.57 ± 0.03pert. ± 0.01αs ± 0.07non−pert. ) GeV
(51)
for the PS mass, with almost no modification compared to the second-order analysis
[34]. Our numerical study of the Coulomb corrections to the top quark pair production cross section near threshold shows that the perturbative approach (mandatory once
non-Coulomb corrections are included) works and led to the conclusion that the residual uncertainty from the Coulomb corrections is less than 5%. This lends support to
the optimistic interpretation that the magnitude of the tt̄ threshold cross section can
eventually be computed with an accuracy of a few percent.
Note added
During the preparation of this paper Penin, Smirnov and Steinhauser [25] have obtained
results for the Coulomb correction to the second and third energy level and wave function
at the origin (n = 2, 3), which agree with our result for general n. We thank the authors
for communicating and comparing their results prior to publication.
Acknowledgements
This work was supported by the DFG Sonderforschungsbereich/Transregio 9 “Computergestützte Theoretische Teilchenphysik”. K.S. acknowledges support of the DFG Graduiertenkolleg “Elementarteilchenphysik an der TeV-Skala”.
19
Appendix
The most difficult part of the third-order calculation of the Green function is the threefold
insertion h0|Ĝ0δV1 Ĝ0 δV1 Ĝ0 δV1 Ĝ0 |0i of the first-order perturbation potential
δ Ṽ1 (q) = −
µ2
4πCF αs αs
a
+
β
ln
.
1
0
q 2 4π
q2
(52)
The matrix element is ultraviolet and infrared finite, so the computation can be done in
three dimensions. Since the external state is a position eigenstate, we Fourier transform
to position space, where the potential has a 1/r and a ln(r)/r term. We can generate
these terms by working with
Wi (ri ) =
ui − 1
1
2
r 2i
4πΓ(1 + 2ui ) cos(πui)
(53)
and by taking the zeroth and first derivative with respect to the ui at ui = 0. The
threefold insertion of the generating potential reads
J3 ≡
=
Z Y
3
d3 ri G0 (0, r1 ; E)W1 (r1 )G0 (r 1 , r 2 ; E)W2 (r2 )G0 (r 2 , r 3 ; E)W3 (r3 )G0 (r 3 , 0; E)
i=1
Z
dr1 dr2 dr3
×
r22u2 +1
r32u3 +1
r12u1 +1
Γ(1 + 2u1 ) cos(πu1 ) Γ(1 + 2u2) cos(πu2) Γ(1 + 2u3 ) cos(πu3 )
(l=0)
(l=0)
G0 (0, r1 , E)G0 (r1 , r2 , E)G0 (r2 , r3 , E)G0 (0, r3 , E)
.
(54)
For the zeroth-order S-wave Coulomb Green functions we use the representations [43, 44]
im2 v imvri
G0 (0, ri , E) = −
e
2π
(l=0)
G0 (ri , rj , E)
Z
∞
2imvri t
dt e
0
1+t
t
λ
,
(55)
∞
(1)
im2 v imv(ri +rj ) X
L(1)
n (−2imvri )Ln (−2imvrj )
=−
e
2π
(n + 1)(n + 1 − λ)
n=0
(56)
q
with v ≡ (E + iǫ)/m and λ = iCF αs /(2v). The Ln(l) (x) are the Laguerre polynomials.
The integrals over ri can now be factorized into two functions H(u, n) and K(u, n, j),
and we obtain
∞ X
∞
m 4X
H(u1 , n)K(u2 , n, j)H(u3, j)
J3 =
,
(57)
4π n=0 j=0 (n + 1)(n + 1 − λ)(j + 1)(j + 1 − λ)
where (defining s = −2imvr)
1
H(u, n) ≡
Γ(1 + 2u) cos πu
K(u, n, j) ≡
eiπ
4m2 v 2
−1
1
Γ(1 + 2u) cos(πu) 4m2 v 2
!u Z
∞
0
1+t
dt
t
eiπ
4m2 v 2
20
!u Z
0
∞
λ Z
0
∞
ds e−(1+t)s s2u+1 L(1)
n (s), (58)
(1)
ds s2u+1e−s L(1)
n (s)Lj (s).
(59)
Performing the integrations in H(u, n) (substituting x = 1/(1 + t) in the t-integral),
we find
Z ∞
1
−u (1 + 2u)(n + 1)
−λ
λ−2−2u
H(u, n) = (−4mE)
dt t (1 + t)
2 F1 −n, 2 + 2u; 2;
cos(πu)
1+t
0
=
n
Γ(2 + k + 2u)Γ(1 + k + 2u)
(−1)k n!
(n + 1)Γ(1 − λ) X
.
u
cos(πu)(−4mE) k=0 k!(n − k)! Γ(1 + 2u)Γ(k + 2)Γ(2 + k + 2u − λ)
(60)
The sum can be expressed in terms of the hypergeometric function 3 F2 (−n, 2 + 2u, 1 +
2u; 2, 2 + 2u − λ; 1), but it is simpler to perform the expansion in the generating variable
u directly. We need the first two terms in the expansion. With the help of the generating
sum
n
X
(−1)k n! Γ(1 + k + a)
Γ(1 + a)Γ(1 − a + n − λ)
=
(61)
k!(n
−
k)!
Γ(2
+
k
−
λ)
Γ(2
+
n
−
λ)Γ(1
−
a
−
λ)
k=0
we find
H(0, n) =
n+1
,
n+1−λ
(62)
H ′ (0, n) = (n + 1)Γ(1 − λ)
n
X
(−1)k n!
Γ(1 + k)
2γE − ln(−4mE)
k=0 k!(n − k)! Γ(2 + k − λ)
+ 2Ψ(1 + k) + 2Ψ(2 + k) − 2Ψ(2 + k − λ)
h
i
n+1
=
− ln(−4mE) − 2 γE + Ψ(n + 1 − λ)
n+1−λ
i
2λ h
Ψ(1 − λ) − Ψ(n + 2 − λ) ,
+
(n + 1)
(63)
where Ψ(z) denotes Euler’s Psi-function, and γE = 0.577216 . . . is Euler’s constant. This
result has already been obtained in [7].
Similarly, for K(u, n, j) we need the expansion up to the first order in u. Here we
obtain
n+1
K(0, n, j) = − 2 2 δnj ,
(64)
4m v
i
1 h
(65)
K ′ (0, n, j) = − 2 2 I + (n + 1) δnj (2γE − ln(−4mE))
4m v
with
I =
Z
0
∞
(1)
ds 2s ln(s)e−s L(1)
n (s)Lj (s).
(66)
To solve the integral I, the Laguerre polynomials are expressed through their generating
functions. Then with
zu
∞
X
e− 1−u
us L(1)
=
s (z),
(1 − u)2 s=0
21
(67)
and
Z
0
∞
ds 2s ln(s)e−s
sw
− 1−w
sv
− 1−v
e
e
=−
2
(1 − v) (1 − w)2
h
2 − 1 + γE + ln
(vw −
1−vw
(v−1)(w−1)
1)2
i
,
(68)
the integral I is expressed as
1−vw
1 1 ∂ n ∂ j  2 −1 + γE + ln (v−1)(w−1)
−
I =
n! j! ∂v n ∂w j
(vw − 1)2

=



 2 + 2(1 + n)Ψ(1 + n)



−2


v=w=0
if n = j
min(n, j) + 1
|j − n|
(69)
if n 6= j
Hence the final result for K ′ (0, n, j) reads
K ′ (0, n, j) = −

n


 2 + (n + 1) 2[γE
1
4m2 v 2 


−2
+ Ψ(1 + n)] − ln(−4mE)
min(n, j) + 1
|j − n|
o
if n = j
if n 6= j
(70)
At this point, we have expressed the threefold insertion of the perturbation potential
in terms of a doubly infinite sum involving Euler’s Psi-function, see (57) . These sums
converge rapidly when the energy argument is evaluated along a line parallel to the real
axis as required in the calculation of the top quark pair production cross section. In
order to obtain the third-order correction to the energy levels and wave functions at
the origin in Section 3, we extract analytically the pole part of J3 when E approaches
the leading-order S-wave bound state energies En(0) , which correspond to λ = n. This
results in multiple sums, which, after a tedious reduction, can all be expressed in terms
of zeta-functions and nested harmonic sums.
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